|
| |
|
|
A108434
|
|
Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
|
|
1
| |
|
|
1, 1, 7, 47, 361, 2977, 25775, 231103, 2127409, 19990241, 190957559, 1848911279, 18104425561, 178975914433, 1783843502047, 17906040994559, 180858717257185, 1836792828317761, 18745545101801063, 192145823547338927
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Column 0 of A108433.
|
|
|
REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
|
|
|
FORMULA
| G.f. = 1/(1+z-zA-zA^2), where A=1+zA^2+zA^3 or, equivalently, A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
|
|
|
EXAMPLE
| a(2)=7 because we have uudd, uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd.
|
|
|
MAPLE
| g:=1/(1+z-z*A-z*A^2): A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3:gser:=series(g, z=0, 27): 1, seq(coeff(gser, z^n), n=1..24);
|
|
|
CROSSREFS
| Cf. A027307, A108431, A108432, A108433.
Sequence in context: A024187 A001711 A088057 * A093173 A173772 A178002
Adjacent sequences: A108431 A108432 A108433 * A108435 A108436 A108437
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 03 2005
|
| |
|
|
